Imo 2010

This is the wiki page for the mini-polymath2 project, which seeks solutions to Question 5 of the 2010 International Mathematical Olympiad.

The project will start at 16:00 UTC July 8, and is hosted at the polymath blog. A discussion thread is hosted at Terry Tao's blog.

Rules

This project will follow the usual polymath rules. In particular:

• Everyone is welcome to participate, though people who have already seen an external solution to the problem should probably refrain from giving spoilers throughout the experiment.
• This is a team effort, not a race between individuals. Rather than work for extended periods of time in isolation from the rest of the project, the idea is to come up with short observations (or to carry an observation of another participant further) and then report back what one gets to the rest of the team. Partial results or even failures can be worth reporting.
• Participants are encouraged to update the wiki, or to summarise progress within threads, for the benefit of others.

Threads

Discussion and planning:

• Future mini-polymath project: 2010 IMO Q6? June 12, 2010.
• Organising mini-polymath2 June 21, 2010.
• Mini-polymath2 start time, June 27, 2010.
• Mini-polymath2 discussion thread, July 8 2010.

Research:

• Minipolymath2 project: IMO 2010 Q5, July 8 2010.

The question

The question to be solved is Question 5 of the 2010 International Mathematical Olympiad:

Problem In each of six boxes [math]\displaystyle{ B_1, B_2, B_3, B_4, B_5, B_6 }[/math] there is initially one coin. There are two types of operation allowed:

Type 1: Choose a nonempty box [math]\displaystyle{ B_j }[/math] with [math]\displaystyle{ 1 \leq j \leq 5 }[/math]. Remove one coin from [math]\displaystyle{ B_j }[/math] and add two coins to [math]\displaystyle{ B_{j+1} }[/math].

Type 2: Choose a nonempty box [math]\displaystyle{ B_k }[/math] with [math]\displaystyle{ 1 \leq k \leq 4 }[/math]. Remove one coin from [math]\displaystyle{ B_k }[/math] and exchange the contents of (possibly empty) boxes [math]\displaystyle{ B_{k+1} }[/math] and [math]\displaystyle{ B_{k+2} }[/math].

Determine whether there is a finite sequence of such operations that results in boxes [math]\displaystyle{ B_1, B_2, B_3, B_4, B_5 }[/math] being empty and box [math]\displaystyle{ B_6 }[/math] containing exactly [math]\displaystyle{ 2010^{2010^{2010}} }[/math] coins. (Note that [math]\displaystyle{ a^{b^c} := a^{(b^c)} }[/math].)

Observations and partial results

• If the left-most box [math]\displaystyle{ B_1 }[/math] becomes empty, then it cannot ever become non-empty again. Furthermore, the left-most box can never have more than one coin; it can be touched exactly once.
• Define the worth W of a state to be [math]\displaystyle{ W = B_6 + 2 B_5 + 4 B_4 + 8 B_3 + 16 B_2 + 32 B_1 }[/math]. Then the initial worth is 63, the final desired worth is [math]\displaystyle{ 2010^{2010^{2010}} }[/math], and the Type 1 move does not affect the worth. On the other hand, the Type 2 move increases the worth when [math]\displaystyle{ B_{j+2} - B_{j+1} \geq 4 }[/math].
• Once one has a large number of coins in one of the first four boxes, say [math]\displaystyle{ B_k }[/math], one can apply the Type 2 move repeatedly to remove coins from [math]\displaystyle{ B_k }[/math] while swapping [math]\displaystyle{ B_{k+1} }[/math] and [math]\displaystyle{ B_{k+2} }[/math] repeatedly. This suggests that it is relatively easy to remove coins from the system; the difficulty is in adding coins to the system.

Possible strategies

• Split the problem into two pieces. Part I: try to show the weaker result that the number of coins in the system can eventually be as large as [math]\displaystyle{ 2010^{2010^{2010}} }[/math]. Part II: Show that once one has a lot of coins, one can move to the final state where [math]\displaystyle{ B_1=\ldots=B_5=0 }[/math] and [math]\displaystyle{ B_6 = 2010^{2010^{2010}} }[/math].
• Try to show that a quantity such as the worth increases or decreases in a controlled manner as one applies the Type 1 and Type 2 moves.
• We know that the first box can never contain more than one coin. What can we say about the second box, third box, etc.?
  • There may be a recursive formula for the maximal size of box [math]\displaystyle{ B_j }[/math], possibly requiring one to solve the five-box, four-box, etc. problems first.
• Work backwards?
• Try to completely solve the three-box problem (say) first: starting from [math]\displaystyle{ [X,Y,Z] }[/math], what is the most number of coins one can generate?

Compound moves

Here we use Type 1 move [math]\displaystyle{ [X,Y] \mapsto [X-1,Y+2] }[/math] and the Type 2 move [math]\displaystyle{ [N,X,Y] \mapsto [N-1,Y,X] }[/math] to create more advanced moves.

1. We can create the move [math]\displaystyle{ [1,X,Y] \mapsto [0,0,X+2Y] }[/math] by applying Type 2 once and then Type 1 Y times.
2. More generally, we have [math]\displaystyle{ [X,Y] \mapsto [0,Y+2X] }[/math] from repeated application of Type 1.
3. We have [math]\displaystyle{ [N,0,0] \to [N-1,2,0] \to [N-1,0,4] \to [N-2,4,0] \to [N-2,0,4] \to \ldots \to [0,0,2^{N+1}] }[/math].

1. [math]\displaystyle{ [a,b,0,0] \to [a,0,2^b,0] \to [a-1,2^b,0,0] \to [a-1,0,2^{2^b},0] \to \cdots }[/math]. So let [math]\displaystyle{ x_0=b, x_{n+1}=2^b }[/math] then we obtain [math]\displaystyle{ [0,x_a,0,0] }[/math], which is huge enough. Making use of this significant compound move, we can obtain very huge number in the fourth coordinate. Then we apply Type 2 move to make the number smaller to fit it to [math]\displaystyle{ 2010^{2010^2010}/4 }[/math], and move it to the sixth coordinate using Type 1 move.

World records

To make the second box as big as possible:

• [math]\displaystyle{ [1,1,1] \mapsto [1,0,3] \mapsto [0,3,0] }[/math] places 3 coins in box 2.

To make the third box as big as possible:

• [math]\displaystyle{ [1,1,1] \mapsto [0,3,1] \mapsto [0,0,7] }[/math] places 7 coins in box 3. (Here we use advanced move 2).

To make the fourth box as big as possible

• [math]\displaystyle{ [1,1,1,1] \to [0,3,0,3] \to [0,2,2,3] \to [0,2,0,7] \to [0,1,7,0] \to [0,1,0,14] }[/math] [math]\displaystyle{ \to [0,0,14,0] \to [0,0,0,28] }[/math] gives 28 coins in box 4.

To make the sixth box as big as possible

• Using the fourth box move we have [math]\displaystyle{ [0,0,0,28,1,1] }[/math]. We can move [math]\displaystyle{ [28,1,1] }[/math] to [math]\displaystyle{ [27,0,7] }[/math] through Type 1; then using a variant of advanced move 3 we get [math]\displaystyle{ [0,0,7 * 2^{27}] }[/math], leading to [math]\displaystyle{ 7 \times 2^{27} }[/math] coins in the last box.